张量积Bezier曲面的降阶

应用张量积Bezier曲面的几何性质和遗传算法，给出了Bezier曲面的降阶。与已有的算法相比，该算法具有计算简单、逼近误差直接给出，几何直观性强等优点。     

正则Bezier曲线的等距线及其计算机实现

利用de Casteljau算法求得正则Bezier曲线上各点处的切矢，再由此得到各点处的法矢，应用于求原始曲线的等距线，该方法几何意义明显，算法简洁。同时给出了用MATLAB绘制Bezier曲线及其等距线的程序，准确快捷，实践效果较好。     

From conics to NURBS: A tutorial and survey

The main geometric features of the nonuniform rational B-splines (NURBS) curve and surface representations are described. It is shown that most of these features are already exhibited by conics, which are a special case of NURBS. The properties typical of NURBS are discussed without dwelling on properties already present in polynomial curves. Conic sections and their representations using rational Bezier curves are reviewed. Cubic NURB curves, geometrical rational splines, rational and B-spline surfaces, and rational Bezier triangles are discussed     

Geometric Properties of Ribs and Fans of a Bezier Curve

Ribs and fans are interesting geometric entities that are derived from a given Béezier curve or surface based on the recent theory of rib and fan decomposition. In this paper, we present some of new geometric properties of ribs and fans for a Béezier curve including composite fans, rib-invariant deformation, and fan-continuity in subdivision. We also give some examples for the presented properties. A preliminary version of this paper appeared in Proc. the 1st Korea-China Joint Conference on Geometric and Visual Computing. Joo-Haeng Lee received his B.S., M.S., and Ph.D. degrees in computer science from POSTECH, Korea, in 1994, 1996 and 1999, respectively. He joined ETRI (Electronics and Telecommunications Research Institute), Korea in 1999, and is currently a senior researcher at Digital Contents Division. His research interests include geometric modeling and processing algorithms for computer graphics, CAD, and robotics. Hyungjun Park is a professor of industrial engineering at Chosun University, Korea. He received his B.S., M.S., and Ph.D. degrees in industrial engineering from POSTECH, Korea, in 1991, 1993, and 1996, respectively. From 1996 to 2001, he worked as a senior researcher at Samsung Electronics, Korea. His current research interests include geometric modeling, computational geometry, reverse engineering, bio-medical engineering, virtual prototyping, and CAD/CAM.     

三次Bezier曲线的一种双参数扩展及应用

对三次Bernstein基函数进行扩展,给出了含有双参数λ,μ的一组四次多项式基函数,基于该组基定义了带双参数的多项式曲线。该曲线不仅具有三次Bezier曲线的诸多特性,而且具有更加灵活的形状可调性。参数λ,μ的几何意义非常明显：在控制顶点不变的情况下,λ,μ分别起到了对曲线相对于控制多边形两内顶点的推拉作用,当λ=μ时,曲线退化为三次Bezier曲线的单参数扩展情形。重点讨论了在不改变控制点位置的情况下如何实现两曲线间的C¹拼接。     

Bezier曲线树

本文提出Ｂｅｚｉｅｒ样条曲线树结构用以高效地实现对曲边几何体的各种几何运算。对任意一条Ｂｅｚｉｅｒ样条曲线，用ｄｅＣａｓｔｅｌｊａｕ算法进行分割后，将分割结果用一平衡二叉树来存储。此后，对于该曲线所作的各种几何运算可以一种逼近等级的方式来进行。先以粗糙的逼近来进行计算，必要时增加精密度，从而提高各种运算的效率。     

Vector elimination: A technique for the implicitization, inversion, and intersection of planar parametric rational polynomial curves

In this paper vector techniques and elimination methods are combined to help resolve some classical problems in computer aided geometric design. Vector techniques are applied to derive the Bezout resultant for two polynomials in one variable. This resultant is then used to solve the following two geometric problems: Given a planar parametric rational polynomial curve, (a) find the implicit polynomial equation of the curve (implicitization); (b) find the parameter value(s) corresponding to the coordinates of a point known to lie on the curve (inversion). The solutions to these two problems are closed form and, in general, require only the arithmetic operations of addition, subtraction, multiplication, and division. These closed form solutions lead to a simple, non-iterative, analytic algorithm for computing the intersection points of two planar parametric rational polynomial curves. Extensions of these techniques to planar rational Bezier curves are also discussed.     

Interactive curve drawing by segmented Bezier approximation with a control parameter

The Bezier approximation technique is considered for curve drawing and curve fitting in computer graphics. Segmented Bezier approximation with a control parameter is proposed for fast and efficient curve approximation. The control parameter is very useful in interactive graphics. Interesting properties of the proposed technique are proved and experimental results are presented. The results are compared with those using the B-spline approximation technique.     

作战标号颜色与几何视觉编码融合显示技术

针对作战装备及野外战场环境的信息呈现,基于作战标号颜色与几何视觉编码的融合显示技术,提出一种对颜色军事语义通过几何辅助进行强化表征的方法,采用贝塞尔曲线对作战标号进行视觉优化处理；设计了“矩形+贝赛尔曲线”的融合显示样式,对作战标号的可视化方式进行研究,使显示样式具有较好的光滑性、视觉分离等效果；增强了作战标号的军事表达能力与人机交互功能,该设计能较好地提升战场态势的信息呈现水平。     

A rational model of the surface swept by a curve*

This paper shows how to construct a rational Bezier model of a swept surface that interpolates N frames (i.e., N position/orientation pairs) of a fixed rational space curve c(s) and maintains the shape of the curve at all intermediate points of the sweep. Thus, the surface models an exact sweep of the curve, consistent with the given data. The primary novelty of the method is that this exact modeling of the sweep is achieved without sacrificing a rational representation for the surface. Through a simple extension, we also allow the sweeping curve to change its size through the sweep. The position, orientation, and size of the sweeping curve can change with arbitrary continuity (we use C² continuity in this paper). Our interpolation between frames has the classical properties of Bezier interpolation, such as the convex hull property and linear precision. This swept surface is a useful primitive for geometric design. It encompasses the surface of revolution and extruded surface, but extends them to arbitrary sweeps. It is a useful modeling primitive for robotics and CAD/CAM, using frames generated automatically by a moving robot or tool.     

保形参数四次插值算法

利用四次的Bézier曲线段构造了GC2连续的参数四次插值样条曲线,该样条曲线是保形的和局部的,且计算十分简单,所有的Bézier点由型值点和曲率直接计算产生,避免了求解矢量方程.最后,给出了一个数值实例.     

由深度数据重建三维物体的一种方法

讨论由深度数据重建三维物体的一种新方法.其基本思想是根据由基于物体旋转的主动 式三角法三维信息获取系统所获取的物体深度数据作为该物体的型值点.再根据型值点进行 网格划分,以便形成满足一定约束条件的空间拓扑网格.最后使用有理Bezier三角曲面片与 矩形曲面片相结合的方法进行曲面拟合与逼近.     

Bézier样条曲线的近似弧长参数化方法

提出了Bézier样条曲线近似弧长参数化的方法及相应的算法。通过求出曲线近似二分之一弧长的点及其相应的参数值,可将曲线分割为两条Bézier样条曲线。这两条曲线的弧长近似相等,因此让它们带有相同的权1。对新生成的Bézier样条曲线不断重复上述工作,最终得到一条由多条Bézier样条曲线所构成的新的曲线。将这多条Bézier样条曲线合并为一条Bézier样条曲线,进而通过节点插入技术将其转化为B样条形式的曲线以便得到全局参数,其中各段Bézier曲线在全局参数域中所占子区间的长度与它们所具有的权成比例,这样便得到一条近似弧长参数化曲线。     

保形五次几何Hermite插值的构造算法

讨论了计算机辅助几何设计中的GHI问题,GHI曲线需要型值点处的切线和曲率信息,所以GHI曲线比一般的插值曲线更困难.首先将保概念引入到GHI曲线,再用分段五次Bezier曲线构造了GC2保形GHI算法.该曲线的所有Bezier点由型值点及相应的曲率信息直接计算产生,无需求解矢量方程组,因此该曲线计算简单,局部修改方便.最后,两个数值例子被给出。     

Bezier分形曲线的细分叠加生成方法

给出了一种由Bezier曲线生成分形曲线的细分叠加方法。将参数二周期化后的Bezier曲线进行递归细分,得到细分曲线序列,再依次将此细分曲线序列无限叠加,构造出处处连续而处处不可微的分形曲线,具有某种自相似性。此Bezier分形曲线可表示为原Bezier曲线控制顶点的线性组合,其调配函数由参数二周期化后的Bernstein基函数无限细分叠加生成,处处连续而处处不可微,且有某种自相似性。数值实验表明此细分叠加方法所生成的曲线具有分形特征。     

Data reduction using cubic rational B-splines

A geometric method for fitting rational cubic B-spline curves to data representing smooth curves, such as intersection curves or silhouette lines, is presented. The algorithm relies on the convex hull and on the variation diminishing properties of Bezier/B-spline curves. It is shown that the algorithm delivers fitting curves that approximate the data with high accuracy even in cases with large tolerances. The ways in which the algorithm computes the end tangent magnitudes and inner control points, fits cubic curves through intermediate points, checks the approximate error, obtains optimal segmentation using binary search, and obtains appropriate final curve form are discussed     

基于曲率分析的三次Bezier曲线采样方法的研究

针对三维建模过程中数据量大的缺点,提出一种简单的基于曲率分析的三次Bezier曲线采样方法。该方法采用每个分段的三次Bezier曲线的特征点和该段曲率半径的极小值作为采样密度的判断标准,曲线采样主要分为多层轮廓和单一轮廓两种情况,对于多层轮廓,采样密度涉及到的因素有曲线特征点,曲率半径极小值,轮廓之间的间距,曲线的长度。而对于单一轮廓,采样密度涉及到的因素有曲线特征点,曲率半径极小值,曲线的长度。通过以上因素,计算出采样点的数目。实验结果证明,提出的方法可行有效,可用于三维建模的数据点采样。     

Low-degree approximation of high-degree B-spline surfaces

In this paper, a method for approximate conversion of high degree Bezier and B-spline surfaces to lower degree representations is presented to facilitate the exchange of surface geometry between different geometric modeling systems. Building on previous work on curve approximation, the method uses adaptive sampling to compute approximation error and lofting of isoparametric curves to produce the approximating surface. In addition, a bound for the approximation accuracy is computed using convex hulls.     

增量极坐标编码的贝赛尔曲线智能优化算法

针对基于统计的隶属度函数确定方法进行了改进,使用贝塞尔曲线作为隶属度函数的上升或下降沿,使隶属度函数可以经过统计结果规定的任意中间点。使用新的增量极坐标编码对贝塞尔曲线控制点进行表达,解决了传统贝塞尔曲线优化中的控制点约束问题。采用差分进化算法对贝塞尔曲线控制点进行优化,可智能拟合经过任意点的最佳贝塞尔曲线。算法可扩展到任意阶贝塞尔曲线,所得隶属度函数较非贝塞尔曲线方法更为合理。